The document discusses odd and even functions. Odd functions satisfy f(-x) = -f(x) and have point symmetry about the origin. Examples given include y = x3 and y = x7 - x5. Even functions satisfy f(-x) = f(x) and have line symmetry about the y-axis. Examples provided are y = x2 and y = x2 + 4. The key characteristics of odd and even functions are outlined and examples are used to prove that certain functions, such as y = x3 + x7, are odd functions.
12X1 T09 08 binomial probability (2010)Nigel Simmons
The document discusses binomial probability distributions. It explains that if an event has two possible outcomes and is repeated, the probability of each outcome follows a binomial distribution. It provides examples of calculating binomial probabilities for 1, 2, 3, and 4 events. The key points are:
- Binomial probabilities use the formula P(X=k) = nCk * pk * (1-p)(n-k)
- This calculates the probability of k successes in n trials with probability of success p
- It works through examples such as drawing balls from a bag to calculate various probabilities
The document discusses motion around a banked curve. It defines the horizontal and vertical forces involved when an object moves around such a curve. It shows that for no sideways force on the object, the horizontal centrifugal force must equal the horizontal component of the normal force from the bank. This allows deriving an equation that relates the curve radius, angle of the bank, and ideal speed to maintain this balance of forces. As an example, it calculates the most favorable speed for a train moving around a banked curve of given radius and rail dimensions.
The document describes a method for solving equations of the form asinx + bcosx = c. It involves using a trigonometric identity to rewrite the equation in terms of the tangent of an angle, letting t = tan(θ/2). This results in a quadratic equation that can be solved for t, and then the inverse tangent gives the solutions for θ. An example problem is worked through step-by-step to demonstrate the method.
The document discusses finding the locus of complex numbers ω or z given some condition on ω or z, where ω = f(z). It provides examples of determining the locus when:
1) ω is purely real or purely imaginary
2) The argument of a linear function of ω or z is equal to an angle θ
3) z satisfies the condition w = (z + 1)/(z - 1) and w is purely real
In the examples, it is shown that the loci are circles, lines, or the real axis depending on the specific condition given. The steps involve making the condition the subject of the equation and then solving to determine the locus.
The document discusses permutations and the basic counting principle. It provides examples of calculating the number of permutations when rolling dice and mice exiting a maze. The key points are:
1) The basic counting principle states that if one event can occur in m ways and another in n ways, the total number of ways the two events can occur together is mn.
2) When rolling 3 dice, there are 6 possibilities for each die, so the total number of permutations is 6 x 6 x 6 = 216.
3) If 4 mice exit a maze through 5 exits independently, the probability of all 4 exiting through the same exit is 1/125.
The document defines and explains key concepts related to polynomial functions. A polynomial P(x) of degree n is an expression of the form P(x) = p0 + p1x + p2x2 + ... + pn-1xn-1 + pnxn, where pn ≠ 0. The degree of a polynomial is the highest exponent in the polynomial. Other important terms defined include coefficients, leading term, leading coefficient, monic polynomials, roots, and zeros. Examples are provided to demonstrate identifying polynomials and determining the degree and form of a given polynomial.
The document discusses odd and even functions. Odd functions satisfy f(-x) = -f(x) and have point symmetry about the origin. Examples given include y = x3 and y = x7 - x5. Even functions satisfy f(-x) = f(x) and have line symmetry about the y-axis. Examples provided are y = x2 and y = x2 + 4. The key characteristics of odd and even functions are outlined and examples are used to prove that certain functions, such as y = x3 + x7, are odd functions.
12X1 T09 08 binomial probability (2010)Nigel Simmons
The document discusses binomial probability distributions. It explains that if an event has two possible outcomes and is repeated, the probability of each outcome follows a binomial distribution. It provides examples of calculating binomial probabilities for 1, 2, 3, and 4 events. The key points are:
- Binomial probabilities use the formula P(X=k) = nCk * pk * (1-p)(n-k)
- This calculates the probability of k successes in n trials with probability of success p
- It works through examples such as drawing balls from a bag to calculate various probabilities
The document discusses motion around a banked curve. It defines the horizontal and vertical forces involved when an object moves around such a curve. It shows that for no sideways force on the object, the horizontal centrifugal force must equal the horizontal component of the normal force from the bank. This allows deriving an equation that relates the curve radius, angle of the bank, and ideal speed to maintain this balance of forces. As an example, it calculates the most favorable speed for a train moving around a banked curve of given radius and rail dimensions.
The document describes a method for solving equations of the form asinx + bcosx = c. It involves using a trigonometric identity to rewrite the equation in terms of the tangent of an angle, letting t = tan(θ/2). This results in a quadratic equation that can be solved for t, and then the inverse tangent gives the solutions for θ. An example problem is worked through step-by-step to demonstrate the method.
The document discusses finding the locus of complex numbers ω or z given some condition on ω or z, where ω = f(z). It provides examples of determining the locus when:
1) ω is purely real or purely imaginary
2) The argument of a linear function of ω or z is equal to an angle θ
3) z satisfies the condition w = (z + 1)/(z - 1) and w is purely real
In the examples, it is shown that the loci are circles, lines, or the real axis depending on the specific condition given. The steps involve making the condition the subject of the equation and then solving to determine the locus.
The document discusses permutations and the basic counting principle. It provides examples of calculating the number of permutations when rolling dice and mice exiting a maze. The key points are:
1) The basic counting principle states that if one event can occur in m ways and another in n ways, the total number of ways the two events can occur together is mn.
2) When rolling 3 dice, there are 6 possibilities for each die, so the total number of permutations is 6 x 6 x 6 = 216.
3) If 4 mice exit a maze through 5 exits independently, the probability of all 4 exiting through the same exit is 1/125.
The document defines and explains key concepts related to polynomial functions. A polynomial P(x) of degree n is an expression of the form P(x) = p0 + p1x + p2x2 + ... + pn-1xn-1 + pnxn, where pn ≠ 0. The degree of a polynomial is the highest exponent in the polynomial. Other important terms defined include coefficients, leading term, leading coefficient, monic polynomials, roots, and zeros. Examples are provided to demonstrate identifying polynomials and determining the degree and form of a given polynomial.
The document defines angular velocity as the rate of change of the angle swept out by a point moving along a circular path with respect to time. It shows that the linear or tangential velocity of the point is equal to the product of its angular velocity and radius. The period of motion is defined as the time taken for one complete revolution, which is calculated by dividing 2π by the angular velocity. An example calculates the angular velocity and tangential velocity of a satellite in circular orbit.
12X1 T07 01 v and a In terms of x (2010)Nigel Simmons
The document discusses the relationship between velocity, acceleration, and position for particles moving in one dimension.
It first shows that if velocity v is a function of position x, the acceleration is equal to the derivative of v squared with respect to x, divided by 2.
It then works through two examples:
1) Finding the velocity of a particle given its acceleration of 3 - 2x as a function of x.
2) Finding the position x of a particle in terms of time t, given its acceleration is 3x^2 and its initial position and velocity.
This document discusses using Cartesian coordinates to find tangents and normals to a parabola with the equation y = x^2/4a.
(1) It shows that the slope of the tangent line at a point P(x1,y1) on the parabola is 2a/x1.
(2) It then derives that the slope of the normal line is -x1/2a.
(3) It analyzes when a line y = mx + b will cut, touch or miss the parabola based on the discriminant Δ = (16a^2)m^2 + 16ab of the system of equations for their intersection. It concludes that
X2 T04 05 curve sketching - powers of functionsNigel Simmons
The document describes how to sketch the graph of y = [f(x)]n, where n is an integer greater than 1. It notes that the graph can be drawn by first sketching y = f(x) and observing that: stationary points and x-intercepts remain the same; [f(x)]n is greater than f(x) if f(x) > 1, and less if f(x) < 1; if n is even, [f(x)]n is always positive; if n is odd, [f(x)]n has the same sign as f(x).
12X1 T09 01 definitions and theory (2010)Nigel Simmons
The document defines key probability terms like probability, sample space, equally likely events, mutually exclusive events, and non-mutually exclusive events. It also provides the formula for calculating probability as the number of favorable outcomes divided by the total number of possible outcomes. As an example, it calculates the probability of throwing a total of 3 or 7 when rolling a pair of dice.
The document discusses calculating tangents to parametric curves using parametrics.
It first shows finding the tangent line at a point P(2ap, ap^2) on the curve x=2at, y=at^2. The slope is shown to be p.
It then shows finding tangents from an external point Q(2q, q^2) to the curve x^2=4y. The slope at the point P(2p, p^2) is p, and the equation of the tangent line is y-p^2=p(x-2p).
The document defines key terms related to quadratic polynomials, functions, and equations. It states that the graph of a quadratic function is a parabola, and discusses how the coefficients and roots of the function determine properties of the parabola like its orientation (concave up or down), intercepts, axis of symmetry, and vertex.
The document discusses the formula for calculating the perpendicular distance from a point to a line. It states that the perpendicular distance is the shortest distance. The formula is given as d = (Ax1 + By1 + C)/√(A2 + B2), where (x1, y1) are the coordinates of the point and Ax + By + C = 0 is the equation of the line. An example is worked through to find the equation of a circle given its tangent line and center point. It also discusses how the sign of (Ax1 + By1 + C) indicates which side of the line a point lies.
The document discusses reduction formulae, which expresses a given integral as the sum of a function and a known integral. It provides an example of using integration by parts to derive the reduction formula for the integral of cos(nx)dx and evaluates the integral of cos(5x)dx. It also shows how to find the reduction formula for the integral of cot(nx)dx and states that this formula can be used to find the value of I6, the integral of cot(6x)dx.
The document discusses the relationships between the roots and coefficients of polynomial equations. It provides formulas to find the sum of roots taken one, two, three, or more at a time in terms of the coefficients. For a polynomial of degree n in the form ax^n + bx^(n-1) + cx^(n-2) + ..., the formulas are provided to find the sum of roots one at a time as -b/a, two at a time as c/a, three at a time as -d/a, and so on. An example is also given to demonstrate using the formulas.
The document discusses mathematical induction and provides two examples of using it to prove statements. It first proves that the expression nn+1nn+2 is divisible by 3 for all positive integers n. It shows the basis step for n=1 and inductive step, assuming true for n=k and proving for n+1. Secondly, it proves 33n+2n+2 is divisible by 5 for all n, again using a basis step and inductive step. The document demonstrates the key steps of mathematical induction.
11X1 T14 05 sum of an arithmetic series (2010)Nigel Simmons
This document discusses how to calculate the sum of an arithmetic series. It states that if the first term (a), the last term (l), and the common difference (d) between terms are known, then the sum can be calculated as (a + l) * n / 2. If only the number of terms (n) and the common difference are known, then the sum can be calculated as n * (the first term + the last term) / 2. The document then provides an example of terms in an arithmetic series.
The document discusses factorizing complex expressions. It states that if a polynomial's coefficients are real, its roots will appear in complex conjugate pairs. Any polynomial of degree n can be factorized into a mixture of quadratic and linear factors over real numbers or into n linear factors over complex numbers. Odd degree polynomials must have at least one real root. Examples of factorizing polynomials over both real and complex numbers are provided.
11 x1 t14 01 definitions & arithmetic series (2012)Nigel Simmons
The document defines arithmetic series as a sequence of numbers where each term is found by adding a constant amount (called the common difference) to the previous term. It provides the general formula for an nth term in an arithmetic series as Tn = a + (n-1)d, where a is the first term and d is the common difference. As an example, it calculates the general term for a series where T3 = 9 and T7 = 21, finding the common difference d = 3 and first term a = 3, giving the formula Tn = 3n - 3. It is then asked to calculate the 100th term T100 for this series.
The document discusses two methods for calculating the area between a curve and the x- or y-axis.
1) For the area below the x-axis (A1), it is given by the integral of the function f(x) between the bounds a and b.
2) For the area on the y-axis between coordinates (a,c) and (b,d), it involves making x the subject of the equation (x=g(y)), then calculating the integral of g(y) between c and d.
Examples are given to demonstrate these methods.
The document discusses tree diagrams and their use in calculating probabilities of outcomes. It provides examples of using tree diagrams to calculate the probability of drawing both a boy's name and a girl's name from a hat containing boys and girls names. It also provides an example of using a tree diagram to calculate the probability that someone buying 5 tickets wins exactly one prize in a raffle with 30 tickets and 2 prizes.
Slideshare is discontinuing its Slidecast feature as of February 28, 2014. Existing Slidecasts will be converted to static presentations without audio by April 30, 2014. The document informs users that new slidecasts can be found on myPlick.com or the author's blog starting in 2014. However, myPlick proved unreliable, so future slidecasts will instead be hosted on the author's YouTube channel.
The document defines angular velocity as the rate of change of the angle swept out by a point moving along a circular path with respect to time. It shows that the linear or tangential velocity of the point is equal to the product of its angular velocity and radius. The period of motion is defined as the time taken for one complete revolution, which is calculated by dividing 2π by the angular velocity. An example calculates the angular velocity and tangential velocity of a satellite in circular orbit.
12X1 T07 01 v and a In terms of x (2010)Nigel Simmons
The document discusses the relationship between velocity, acceleration, and position for particles moving in one dimension.
It first shows that if velocity v is a function of position x, the acceleration is equal to the derivative of v squared with respect to x, divided by 2.
It then works through two examples:
1) Finding the velocity of a particle given its acceleration of 3 - 2x as a function of x.
2) Finding the position x of a particle in terms of time t, given its acceleration is 3x^2 and its initial position and velocity.
This document discusses using Cartesian coordinates to find tangents and normals to a parabola with the equation y = x^2/4a.
(1) It shows that the slope of the tangent line at a point P(x1,y1) on the parabola is 2a/x1.
(2) It then derives that the slope of the normal line is -x1/2a.
(3) It analyzes when a line y = mx + b will cut, touch or miss the parabola based on the discriminant Δ = (16a^2)m^2 + 16ab of the system of equations for their intersection. It concludes that
X2 T04 05 curve sketching - powers of functionsNigel Simmons
The document describes how to sketch the graph of y = [f(x)]n, where n is an integer greater than 1. It notes that the graph can be drawn by first sketching y = f(x) and observing that: stationary points and x-intercepts remain the same; [f(x)]n is greater than f(x) if f(x) > 1, and less if f(x) < 1; if n is even, [f(x)]n is always positive; if n is odd, [f(x)]n has the same sign as f(x).
12X1 T09 01 definitions and theory (2010)Nigel Simmons
The document defines key probability terms like probability, sample space, equally likely events, mutually exclusive events, and non-mutually exclusive events. It also provides the formula for calculating probability as the number of favorable outcomes divided by the total number of possible outcomes. As an example, it calculates the probability of throwing a total of 3 or 7 when rolling a pair of dice.
The document discusses calculating tangents to parametric curves using parametrics.
It first shows finding the tangent line at a point P(2ap, ap^2) on the curve x=2at, y=at^2. The slope is shown to be p.
It then shows finding tangents from an external point Q(2q, q^2) to the curve x^2=4y. The slope at the point P(2p, p^2) is p, and the equation of the tangent line is y-p^2=p(x-2p).
The document defines key terms related to quadratic polynomials, functions, and equations. It states that the graph of a quadratic function is a parabola, and discusses how the coefficients and roots of the function determine properties of the parabola like its orientation (concave up or down), intercepts, axis of symmetry, and vertex.
The document discusses the formula for calculating the perpendicular distance from a point to a line. It states that the perpendicular distance is the shortest distance. The formula is given as d = (Ax1 + By1 + C)/√(A2 + B2), where (x1, y1) are the coordinates of the point and Ax + By + C = 0 is the equation of the line. An example is worked through to find the equation of a circle given its tangent line and center point. It also discusses how the sign of (Ax1 + By1 + C) indicates which side of the line a point lies.
The document discusses reduction formulae, which expresses a given integral as the sum of a function and a known integral. It provides an example of using integration by parts to derive the reduction formula for the integral of cos(nx)dx and evaluates the integral of cos(5x)dx. It also shows how to find the reduction formula for the integral of cot(nx)dx and states that this formula can be used to find the value of I6, the integral of cot(6x)dx.
The document discusses the relationships between the roots and coefficients of polynomial equations. It provides formulas to find the sum of roots taken one, two, three, or more at a time in terms of the coefficients. For a polynomial of degree n in the form ax^n + bx^(n-1) + cx^(n-2) + ..., the formulas are provided to find the sum of roots one at a time as -b/a, two at a time as c/a, three at a time as -d/a, and so on. An example is also given to demonstrate using the formulas.
The document discusses mathematical induction and provides two examples of using it to prove statements. It first proves that the expression nn+1nn+2 is divisible by 3 for all positive integers n. It shows the basis step for n=1 and inductive step, assuming true for n=k and proving for n+1. Secondly, it proves 33n+2n+2 is divisible by 5 for all n, again using a basis step and inductive step. The document demonstrates the key steps of mathematical induction.
11X1 T14 05 sum of an arithmetic series (2010)Nigel Simmons
This document discusses how to calculate the sum of an arithmetic series. It states that if the first term (a), the last term (l), and the common difference (d) between terms are known, then the sum can be calculated as (a + l) * n / 2. If only the number of terms (n) and the common difference are known, then the sum can be calculated as n * (the first term + the last term) / 2. The document then provides an example of terms in an arithmetic series.
The document discusses factorizing complex expressions. It states that if a polynomial's coefficients are real, its roots will appear in complex conjugate pairs. Any polynomial of degree n can be factorized into a mixture of quadratic and linear factors over real numbers or into n linear factors over complex numbers. Odd degree polynomials must have at least one real root. Examples of factorizing polynomials over both real and complex numbers are provided.
11 x1 t14 01 definitions & arithmetic series (2012)Nigel Simmons
The document defines arithmetic series as a sequence of numbers where each term is found by adding a constant amount (called the common difference) to the previous term. It provides the general formula for an nth term in an arithmetic series as Tn = a + (n-1)d, where a is the first term and d is the common difference. As an example, it calculates the general term for a series where T3 = 9 and T7 = 21, finding the common difference d = 3 and first term a = 3, giving the formula Tn = 3n - 3. It is then asked to calculate the 100th term T100 for this series.
The document discusses two methods for calculating the area between a curve and the x- or y-axis.
1) For the area below the x-axis (A1), it is given by the integral of the function f(x) between the bounds a and b.
2) For the area on the y-axis between coordinates (a,c) and (b,d), it involves making x the subject of the equation (x=g(y)), then calculating the integral of g(y) between c and d.
Examples are given to demonstrate these methods.
The document discusses tree diagrams and their use in calculating probabilities of outcomes. It provides examples of using tree diagrams to calculate the probability of drawing both a boy's name and a girl's name from a hat containing boys and girls names. It also provides an example of using a tree diagram to calculate the probability that someone buying 5 tickets wins exactly one prize in a raffle with 30 tickets and 2 prizes.
Slideshare is discontinuing its Slidecast feature as of February 28, 2014. Existing Slidecasts will be converted to static presentations without audio by April 30, 2014. The document informs users that new slidecasts can be found on myPlick.com or the author's blog starting in 2014. However, myPlick proved unreliable, so future slidecasts will instead be hosted on the author's YouTube channel.
The document discusses different methods for factorising expressions:
1) Looking for a common factor and dividing it out of all terms
2) Using the difference of two squares formula (a2 - b2 = (a - b)(a + b))
3) Factorising quadratic trinomials into two binomial factors by identifying the values that multiply to give the constant term and sum to give the coefficient of the linear term.
The document provides information on index laws and the meaning of indices in algebra:
- Index laws state that am × an = am+n, am ÷ an = am-n, and (am)n = amn. Exponents can be added or subtracted when multiplying or dividing terms with the same base.
- Positive exponents indicate a term is raised to a power. Negative exponents indicate a root is being taken. Terms with exponents are evaluated from left to right.
- Examples demonstrate how to simplify expressions using index laws and interpret different types of indices.
12 x1 t01 03 integrating derivative on function (2013)Nigel Simmons
The document discusses integrating derivatives of functions. It states that the integral of the derivative of a function f(x) is equal to the natural log of f(x) plus a constant. It then provides examples of integrating several derivatives: (i) ∫(1/(7-3x)) dx = -1/3 log(7-3x) + c, (ii) ∫(1/(8x+5)) dx = 1/8 log(8x+5) + c, and (iii) ∫(x5/(x-2)) dx = 1/6 log(x6-2) + c. It also discusses techniques for integrating fractions by polynomial long division and finds
The document discusses logarithms and their properties. Logarithms are defined as the inverse of exponentials. If y = ax, then x = loga y. The natural logarithm is log base e, written as ln. Properties of logarithms include: loga m + loga n = loga mn; loga m - loga n = loga(m/n); loga mn = n loga m; loga 1 = 0; loga a = 1. Examples of evaluating logarithmic expressions are provided.
The document discusses relationships between the coefficients and roots of polynomials. It states that for a polynomial P(x) = axn + bxn-1 + cxn-2 + ..., the sum of the roots equals -b/a, the sum of the roots taken two at a time equals c/a, and so on for higher order terms. It also provides examples of using these relationships to find the sums of roots for a given polynomial.
P
4
3
2
The document discusses properties of polynomials with multiple roots. It first proves that if a polynomial P(x) has a root x = a of multiplicity m, then the derivative of P(x), P'(x), will have a root x = a of multiplicity m-1. It then provides an example of solving a cubic equation given it has a double root. Finally, it examines a quartic polynomial and shows that its root α cannot be 0, 1, or -1, and that 1/
The document discusses factorizing complex expressions. The main points are:
- If a polynomial's coefficients are real, its roots will appear in complex conjugate pairs.
- Any polynomial of degree n can be factorized into a mixture of quadratic and linear factors over real numbers, or into n linear factors over complex numbers.
- Odd degree polynomials must have at least one real root.
- Examples of factorizing polynomials over both real and complex numbers are provided.
The document describes the Trapezoidal Rule for approximating the area under a curve between two points. It shows that the area A is estimated by dividing the region into trapezoids with height equal to the function values at the interval endpoints and bases equal to the intervals. In general, the area is approximated as the sum of the areas of each trapezoid, which is equal to the average of the endpoint function values multiplied by the interval length.
The document discusses methods for calculating the volumes of solids of revolution. It provides formulas for finding volumes when an area is revolved around either the x-axis or y-axis. Examples are given for finding volumes of common solids like cones, spheres, and others. Steps are shown for using the formulas to calculate volumes based on given functions and limits of revolution.
The document discusses different methods for calculating the area under a curve or between curves.
(1) The area below the x-axis is given by the integral of the function between the bounds, which can be positive or negative depending on whether the area is above or below the x-axis.
(2) To calculate the area on the y-axis, the function is solved for x in terms of y, then the bounds are substituted into the integral of this new function with respect to y.
(3) The area between two curves is calculated by taking the integral of the upper curve minus the integral of the lower curve, both between the same bounds on the x-axis.
The document discusses 8 properties of definite integrals:
1) Integrating polynomials results in a fraction.
2) Constants can be factored out of integrals.
3) Integrals of sums are equal to the sum of integrals.
4) Splitting an integral range results in the sum of the integrals.
5) Integrals of positive functions over a range are positive, and negative if the function is negative.
6) Integrals can be compared based on the relative values of the integrands.
7) Changing the limits of integration flips the sign of the integral.
8) Integrals of odd functions over a symmetric range are zero, and integrals of even functions are twice the integral over
6. Sum & Difference of Angles
y
P cos ,sin
1 y
1
x
x x
cos
1
x cos
7. Sum & Difference of Angles
y
P cos ,sin
1 y
1
x
x x y
cos sin
1 1
x cos y sin
8. Sum & Difference of Angles
y
Q cos ,sin P cos ,sin
1 y
1
x
x x y
cos sin
1 1
x cos y sin
9. Sum & Difference of Angles
y
Q cos ,sin P cos ,sin
1 y
1
x
x x y
cos sin
1 1
x cos y sin
By trigonometry
10. Sum & Difference of Angles
y
Q cos ,sin P cos ,sin
1 y
1
x
x x y
cos sin
1 1
x cos y sin
By trigonometry
PQ 2 12 12 2 11 cos
11. Sum & Difference of Angles
y
Q cos ,sin P cos ,sin
1 y
1
x
x x y
cos sin
1 1
x cos y sin
By trigonometry
PQ 2 12 12 2 11 cos
PQ 2 2 2cos
12. Sum & Difference of Angles
y
Q cos ,sin P cos ,sin
1 y
1
x
x x y
cos sin
1 1
x cos y sin
By trigonometry
PQ 2 12 12 2 11 cos
PQ 2 2 2cos
By coordinate geometry
13. Sum & Difference of Angles
y
Q cos ,sin P cos ,sin
1 y
1
x
x x y
cos sin
1 1
x cos y sin
By trigonometry
PQ 2 12 12 2 11 cos
PQ 2 2 2cos
By coordinate geometry
PQ 2 cos cos sin sin
2 2
14. Sum & Difference of Angles
y
Q cos ,sin P cos ,sin
1 y
1
x
x x y
cos sin
1 1
x cos y sin
By trigonometry
PQ 2 12 12 2 11 cos
PQ 2 2 2cos
By coordinate geometry
PQ 2 cos cos sin sin
2 2
PQ 2 cos 2 2cos cos cos 2 sin 2 2sin sin sin 2
15. Sum & Difference of Angles
y
Q cos ,sin P cos ,sin
1 y
1
x
x x y
cos sin
1 1
x cos y sin
By trigonometry
PQ 2 12 12 2 11 cos
PQ 2 2 2cos
By coordinate geometry
PQ 2 cos cos sin sin
2 2
PQ 2 cos 2 2cos cos cos 2 sin 2 2sin sin sin 2
PQ 2 2 2cos cos 2sin sin
17. 2 2cos 2 2cos cos 2sin sin
cos cos cos sin sin
18. 2 2cos 2 2cos cos 2sin sin
cos cos cos sin sin
Replace with
19. 2 2cos 2 2cos cos 2sin sin
cos cos cos sin sin
Replace with
cos cos cos sin sin
20. 2 2cos 2 2cos cos 2sin sin
cos cos cos sin sin
Replace with
cos cos cos sin sin
cos cos (even function i.e. f x f x )
21. 2 2cos 2 2cos cos 2sin sin
cos cos cos sin sin
Replace with
cos cos cos sin sin
cos cos (even function i.e. f x f x )
sin sin (odd function i.e. f x f x )
22. 2 2cos 2 2cos cos 2sin sin
cos cos cos sin sin
Replace with
cos cos cos sin sin
cos cos (even function i.e. f x f x )
sin sin (odd function i.e. f x f x )
cos cos cos sin sin
23. 2 2cos 2 2cos cos 2sin sin
cos cos cos sin sin
Replace with
cos cos cos sin sin
cos cos (even function i.e. f x f x )
sin sin (odd function i.e. f x f x )
cos cos cos sin sin
cos cos cos sin sin
24. 2 2cos 2 2cos cos 2sin sin
cos cos cos sin sin
Replace with
cos cos cos sin sin
cos cos (even function i.e. f x f x )
sin sin (odd function i.e. f x f x )
cos cos cos sin sin
cos, cos, sin, sin
cos cos cos sin sin
25. 2 2cos 2 2cos cos 2sin sin
cos cos cos sin sin
Replace with
cos cos cos sin sin
cos cos (even function i.e. f x f x )
sin sin (odd function i.e. f x f x )
cos cos cos sin sin
cos, cos, sin, sin
cos cos cos sin sin
26. 2 2cos 2 2cos cos 2sin sin
cos cos cos sin sin
Replace with
cos cos cos sin sin
cos cos (even function i.e. f x f x )
sin sin (odd function i.e. f x f x )
cos cos cos sin sin
cos, cos, sin, sin
cos cos cos sin sin
27. 2 2cos 2 2cos cos 2sin sin
cos cos cos sin sin
Replace with
cos cos cos sin sin
cos cos (even function i.e. f x f x )
sin sin (odd function i.e. f x f x )
cos cos cos sin sin
cos, cos, sin, sin
cos cos cos sin sin
28. 2 2cos 2 2cos cos 2sin sin
cos cos cos sin sin
Replace with
cos cos cos sin sin
cos cos (even function i.e. f x f x )
sin sin (odd function i.e. f x f x )
cos cos cos sin sin
cos, cos, sin, sin
cos cos cos sin sin
29. 2 2cos 2 2cos cos 2sin sin
cos cos cos sin sin
Replace with
cos cos cos sin sin
cos cos (even function i.e. f x f x )
sin sin (odd function i.e. f x f x )
cos cos cos sin sin
cos, cos, sin, sin
cos cos cos sin sin
If it’s not the sine
30. 2 2cos 2 2cos cos 2sin sin
cos cos cos sin sin
Replace with
cos cos cos sin sin
cos cos (even function i.e. f x f x )
sin sin (odd function i.e. f x f x )
cos cos cos sin sin
cos, cos, sin, sin
cos cos cos sin sin
If it’s not the sine,it’s not the sign
32. Replace with 90
cos 90 cos 90 cos sin 90 sin
33. Replace with 90
cos 90 cos 90 cos sin 90 sin
cos 90 cos 90 cos sin 90 sin
34. Replace with 90
cos 90 cos 90 cos sin 90 sin
cos 90 cos 90 cos sin 90 sin
sin sin cos cos sin
35. Replace with 90
cos 90 cos 90 cos sin 90 sin
cos 90 cos 90 cos sin 90 sin
sin sin cos cos sin
Replace with
36. Replace with 90
cos 90 cos 90 cos sin 90 sin
cos 90 cos 90 cos sin 90 sin
sin sin cos cos sin
Replace with
sin sin cos cos sin
37. Replace with 90
cos 90 cos 90 cos sin 90 sin
cos 90 cos 90 cos sin 90 sin
sin sin cos cos sin
Replace with
sin sin cos cos sin
sin sin cos cos sin
38. Replace with 90
cos 90 cos 90 cos sin 90 sin
cos 90 cos 90 cos sin 90 sin
sin sin cos cos sin
Replace with
sin sin cos cos sin
sin sin cos cos sin
sin sin cos cos sin
39. Replace with 90
cos 90 cos 90 cos sin 90 sin
cos 90 cos 90 cos sin 90 sin
sin sin cos cos sin
Replace with
sin sin cos cos sin
sin sin cos cos sin
sin, cos, cos, sin
sin sin cos cos sin
40. Replace with 90
cos 90 cos 90 cos sin 90 sin
cos 90 cos 90 cos sin 90 sin
sin sin cos cos sin
Replace with
sin sin cos cos sin
sin sin cos cos sin
sin, cos, cos, sin
sin sin cos cos sin
41. Replace with 90
cos 90 cos 90 cos sin 90 sin
cos 90 cos 90 cos sin 90 sin
sin sin cos cos sin
Replace with
sin sin cos cos sin
sin sin cos cos sin
sin, cos, cos, sin
sin sin cos cos sin
42. Replace with 90
cos 90 cos 90 cos sin 90 sin
cos 90 cos 90 cos sin 90 sin
sin sin cos cos sin
Replace with
sin sin cos cos sin
sin sin cos cos sin
sin, cos, cos, sin
sin sin cos cos sin
43. Replace with 90
cos 90 cos 90 cos sin 90 sin
cos 90 cos 90 cos sin 90 sin
sin sin cos cos sin
Replace with
sin sin cos cos sin
sin sin cos cos sin
sin, cos, cos, sin
sin sin cos cos sin
44. Replace with 90
cos 90 cos 90 cos sin 90 sin
cos 90 cos 90 cos sin 90 sin
sin sin cos cos sin
Replace with
sin sin cos cos sin
sin sin cos cos sin
sin, cos, cos, sin
sin sin cos cos sin
If it’s the sine
45. Replace with 90
cos 90 cos 90 cos sin 90 sin
cos 90 cos 90 cos sin 90 sin
sin sin cos cos sin
Replace with
sin sin cos cos sin
sin sin cos cos sin
sin, cos, cos, sin
sin sin cos cos sin
If it’s the sine,it’s the sign
47. tan
sin
tan
cos
48. tan
sin
tan
cos
sin cos cos sin
tan
cos cos sin sin
49. tan
sin
tan
cos
sin cos cos sin
tan
cos cos sin sin
sin cos cos sin
cos cos cos cos
tan
cos cos sin sin
cos cos cos cos
50. tan
sin
tan
cos
sin cos cos sin
tan
cos cos sin sin
sin cos cos sin
cos cos cos cos
tan
cos cos sin sin
cos cos cos cos
tan tan
tan
1 tan tan
51. tan
sin
tan
cos
sin cos cos sin
tan
cos cos sin sin
sin cos cos sin
cos cos cos cos
tan
cos cos sin sin
cos cos cos cos
tan tan
tan
1 tan tan
Replace with
52. tan
sin
tan
cos
sin cos cos sin
tan
cos cos sin sin
sin cos cos sin
cos cos cos cos
tan
cos cos sin sin
cos cos cos cos
tan tan
tan
1 tan tan
Replace with
tan tan
tan
1 tan tan
53. tan
sin
tan
cos
sin cos cos sin
tan
cos cos sin sin
sin cos cos sin
cos cos cos cos
tan
cos cos sin sin
cos cos cos cos
tan tan
tan
1 tan tan
Replace with
tan tan
tan
1 tan tan tan tan
(odd function i.e. f x f x )
54. tan
sin
tan
cos
sin cos cos sin
tan
cos cos sin sin
sin cos cos sin
cos cos cos cos
tan
cos cos sin sin
cos cos cos cos
tan tan
tan
1 tan tan
Replace with
tan tan
tan
1 tan tan tan tan
tan tan (odd function i.e. f x f x )
tan
1 tan tan
56. tan plus tan
tan tan
tan
1 tan tan
57. tan plus tan
tan tan
tan
1 tan tan
on one minus tan, tan
58. tan plus tan
tan tan
tan
1 tan tan
on one minus tan, tan
Sum and Difference of Angles
59. tan plus tan
tan tan
tan
1 tan tan
on one minus tan, tan
Sum and Difference of Angles
sin sin cos cos sin
60. tan plus tan
tan tan
tan
1 tan tan
on one minus tan, tan
Sum and Difference of Angles
sin sin cos cos sin
cos cos cos sin sin
61. tan plus tan
tan tan
tan
1 tan tan
on one minus tan, tan
Sum and Difference of Angles
sin sin cos cos sin
cos cos cos sin sin
tan tan
tan
1 tan tan
63. e.g. i Expand cos 2 3
cos 2 3 cos 2 cos3 sin 2 sin 3
64. e.g. i Expand cos 2 3
cos 2 3 cos 2 cos3 sin 2 sin 3
tan 20 tan10
(ii) Simplify
1 tan 20 tan10
65. e.g. i Expand cos 2 3
cos 2 3 cos 2 cos3 sin 2 sin 3
tan 20 tan10
(ii) Simplify
1 tan 20 tan10
tan 20 tan10
tan 20 10
1 tan 20 tan10
66. e.g. i Expand cos 2 3
cos 2 3 cos 2 cos3 sin 2 sin 3
tan 20 tan10
(ii) Simplify
1 tan 20 tan10
tan 20 tan10
tan 20 10
1 tan 20 tan10
tan 30
67. e.g. i Expand cos 2 3
cos 2 3 cos 2 cos3 sin 2 sin 3
tan 20 tan10
(ii) Simplify
1 tan 20 tan10
tan 20 tan10
tan 20 10
1 tan 20 tan10
tan 30
1
3
68. e.g. i Expand cos 2 3
cos 2 3 cos 2 cos3 sin 2 sin 3
tan 20 tan10
(ii) Simplify
1 tan 20 tan10
tan 20 tan10
tan 20 10
1 tan 20 tan10
tan 30
1
3
(iii) Find the exact value of sin15
69. e.g. i Expand cos 2 3
cos 2 3 cos 2 cos3 sin 2 sin 3
tan 20 tan10
(ii) Simplify
1 tan 20 tan10
tan 20 tan10
tan 20 10
1 tan 20 tan10
tan 30
1
3
(iii) Find the exact value of sin15
sin15 sin 45 30
70. e.g. i Expand cos 2 3
cos 2 3 cos 2 cos3 sin 2 sin 3
tan 20 tan10
(ii) Simplify
1 tan 20 tan10
tan 20 tan10
tan 20 10
1 tan 20 tan10
tan 30
1
3
(iii) Find the exact value of sin15
sin15 sin 45 30
sin 45 cos30 cos 45 sin 30
71. e.g. i Expand cos 2 3
cos 2 3 cos 2 cos3 sin 2 sin 3
tan 20 tan10
(ii) Simplify
1 tan 20 tan10
tan 20 tan10
tan 20 10
1 tan 20 tan10
tan 30
1
3
(iii) Find the exact value of sin15
sin15 sin 45 30
sin 45 cos30 cos 45 sin 30
1 3 1 1
2 2 2 2
72. e.g. i Expand cos 2 3
cos 2 3 cos 2 cos3 sin 2 sin 3
tan 20 tan10
(ii) Simplify
1 tan 20 tan10
tan 20 tan10
tan 20 10
1 tan 20 tan10
tan 30
1
3
(iii) Find the exact value of sin15
sin15 sin 45 30
sin 45 cos30 cos 45 sin 30
1 3 1 1
2 2 2 2
3 1
2 2
73. 2 1
iv If sin and cos , find sin
3 4
74. 2 1
iv If sin and cos , find sin
3 4
3 2
5
75. 2 1
iv If sin and cos , find sin
3 4
3 2 4 15
5 1
76. 2 1
iv If sin and cos , find sin
3 4
3 2 4 15
5 1
sin sin cos cos sin
77. 2 1
iv If sin and cos , find sin
3 4
3 2 4 15
5 1
sin sin cos cos sin
2 1 5 15
3 4 3 4
78. 2 1
iv If sin and cos , find sin
3 4
3 2 4 15
5 1
sin sin cos cos sin
2 1 5 15
3 4 3 4
25 3
12
79. 2 1
iv If sin and cos , find sin
3 4
3 2 4 15
5 1
sin sin cos cos sin
2 1 5 15
3 4 3 4
25 3
12
Exercise 14D; 1ade, 2bce, 5ac, 7, 9ac, 10ac, 12, 13ac, 16ab, 17, 23
